Bill, if the tree was being removed, then growth from this spring, let's say May/June should have been among what was being removed. That growth should be plenty hard enough to survive the winter in your fridge. If you just cut the tips off, then yes, you might have some issues, but if you cut stuff close to the ground, you should be fine.
Also, like someone else said, if it was being completely removed, if you could get a chunk of the roots with a branch/trunk attached, you could just put that in a pot and keep it in your garage for the winter and next spring you'd be good to go.
You will only get gnats if they are introduced somehow, often from the potting mix. If you use a good potting mix you shouldn't have any problems. Maybe root 1 or 2 now to increase your odds?
Is this variety worth keeping in the first place? Do you know what it is? Why not just start fresh with the varieties you want. Unless you could get the whole rootball with some trunk attached, that would give you a headstart on things.